Cylindrical Localized Approximation to Speed up Computations for Gaussian Beams in the Generalized...

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Cylindrical localized approximation to speed upcomputations for Gaussian beams in the generalizedLorenz–Mie theory for cylinders, with arbitrary locationand orientation of the scatterer

Gerard Gouesbet, Kuan Fang Ren, Loic Mees, and Gerard Grehan

A cylindrical localized approximation to speed up numerical computations in generalized Lorenz–Mietheory for cylinders, in a special case of perpendicular illumination, was recently introduced and rigor-ously justified. We generalize this approximation to the case when the cylinder is arbitrarily located andarbitrarily oriented in a Gaussian beam. © 1999 Optical Society of America

OCIS codes: 290.0290, 290.4020.

1. Introduction

The general theory of interaction between an arbi-trarily shaped beam and an infinite cylinder was re-cently developed ~Ref. 1 and references therein!, with,expectedly, many potential applications such as forthe design and analysis of phase Doppler instru-ments.2 In its more general version, the theory hasto be expressed in terms of distributions rather thanin terms of the usual functions, in particular if theilluminating beam is described in the so-called Davisformalism.3,4 This fact results from the status of theeparability theorem in cylindrical coordinates.5By analogy to a similar theory, which is called the

generalized Lorenz–Mie theory ~GLMT! when thecatterers are spheres,6,7 the above theory can be

called the GLMT for cylinders. However, referringto the previous treatment of the interaction betweeninfinite cylinders and plane waves, it may also becalled the generalized Wait theory.8 When it is ex-pressed explicitly in terms of distributions, we say

G. Gouesbet, L. Mees, and G. Grehan are with the Laboratoired’Energetique des Systemes et Procedes, Unite Mixte de Recher-he, 6614 Complex de Recherche Interprofessionnel en Aerother-ochimie, Centre National de la Recherche Scientifique,niversite et Institut National des Sciences Appliques de Rouen,.P. 08, 76131 Mont-Saint-Aignan, France. The e-mail address

or G. Grehan is [email protected]. K. F. Ren is with theadio Physic Laboratory, Xidian University, Xian, China.Received 15 December 1997; revised manuscript received 1 De-

ember 1998.0003-6935y99y122647-19$15.00y0© 1999 Optical Society of America

that the GLMT for cylinders is expressed in the dis-tributive approach.

Another recent study of the interaction betweenarbitrarily shaped beams and infinite cylinders hasbeen published by Lock9,10 and relies on a plane-wavespectrum approach. In the research reported inRefs. 9 and 10, distributions were avoided by choice ofa special parameterization to describe the illuminat-ing beam. However, it has been shown that the dis-tributive and the plane-wave spectrum approachesare equivalent11 and that, in its more general version,the plane-wave spectrum approach also may have todeal with distributions.

The plane-wave spectrum approach, however, is insuitable form for developing the so-called cylindricallocalized approximation. This cylindrical localizedapproximation is akin, in its spirit, to the localizedapproximation that was previously developed for theGLMT for spheres12–15 and definitely permitted theuse of this GLMT for many applications in manyfields, such as in physics, chemistry, and engineering.

The present cylindrical localized approximationwas introduced, in analogy with the localized approx-imation for spheres but without a firm mathematicalbasis, for a special case of perpendicular illumina-tion.11 It was afterward rigorously justified.16 Weremark that, although the cylindrical localized ap-proximation does not rely on distributions, distribu-tions were required for this justification. This paperis devoted to the much more difficult case of arbitrarylocation and arbitrary orientation of the cylinder.Although we consider only Gaussian beams, our pro-cedure is expected to possess a high level of generality

20 April 1999 y Vol. 38, No. 12 y APPLIED OPTICS 2647

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and would likely be applicable to other kinds of beamtoo.

Let us remark that a cylindrical localized approx-imation was also introduced by Lock.9,10 Lock’s lo-calized approximation is based on an angularspectrum of plane-wave modeling of the beam,whereas ours is based on Davis modeling, so the twoapproximations seemingly differ from each other and,at least, are complementary. We suspect, however,that in some sense they might be equivalent. Thisidea arises from the facts that our approximationreceived a rigorous justification and that both approx-imations are efficient. It is then likely that a rigor-ous justification of Lock’s approximation in the samespirit as in the present paper would actually revealits close connection with ours. An investigation ofthis issue is, however, outside the scope of the presentpaper.

Also, to help the reader to implement the localizedapproximation we mention here the essential resultsto be used for Gaussian beams. Beam shape coeffi-cients in the localized approximation are evaluatedby use of Eqs. ~34! and ~35! below, in which the pref-actor K is given by Eq. ~45!. The quantities Ez0

i andHz0

i are defined by Eq. ~32!, in which the localizationoperator G is given by the prescriptions of Eqs. ~58!and ~60!. For the case of perpendicular polarizationand off-axis normal incidence, analytical expressionscan be obtained from Eqs. ~103!, ~104!, ~106!, and~108!, with U0, V0, and W0 defined in Eq. ~18! and

ith u90, v90, and w90 defined in Subsection 3.A. Forthe most general case of arbitrary orientation of thescatterer, we provide analytical truncated expres-sions in Subsection 5.C. Unfortunately, these ex-pressions contain terms that are stored in symboliccomputation files that the reader may have to recon-stitute. A more expedient implementation methodis then to use the quadrature expressions of Eqs. ~34!and ~35!.

This paper is organized as follows: In Section 2we recall some basic features of the GLMT for cylin-ders and, in particular, the relationship between thedistributive and the plane-wave spectrum ap-proaches. In Section 3 we describe the case understudy, introduce the cylindrical localized approxima-tion, and rigorously justify it. In Section 4 we dem-onstrate that, although it is based on a first-orderdescription of the Gaussian beam, the localized ap-proximation anticipates the rigorous formulationthat arises from the use of higher-order beams. InSection 5 we establish algebraic expressions to han-dle the approximation, and in Section 6 we providenumerical results. Section 7 is a conclusion.

2. Basic Features of the Generalized Lorenz–MieTheory for Cylinders

A. Distributive Approach

The theory describes the illuminating beam in termsof two Bromwich scalar potentials UTM

i and UTEi ~i

648 APPLIED OPTICS y Vol. 38, No. 12 y 20 April 1999

means illuminating, TM is transverse magnetic, andTE is transverse electric! that read as1

UTMi 5

E0

k2 (m52`

1`

~2i!m exp~imw!

3 Îm,TM~g!, Jm~RÎ1 2 g2!exp~igZ!&, (1)

UTEi 5

H0

k2 (m52`

1`

~2i!m exp~imw!

3 Îm,TE~g!, Jm~RÎ1 2 g2!exp~igZ!&, (2)

in which Z and R are rescaled cylindrical coordinates:

Z 5 kz, R 5 kr, (3)

where k is the wave number of the illuminatingbeam, z and r are cylindrical coordinates ~Fig. 1!, E0and H0 are the electric and the magnetic strengths ofthe beam, respectively, Jm designates Bessel func-tions of the first kind, Im,TM~g! and Im,TE~g! are theso-called beam-shape distributions ~BSD’s! that de-pend on separation constant g, Â, B& is the standardnotation nowadays for using distributions,17–19 quan-ity A is the distribution itself, and quantity B is a

so-called test function. The test functions in Eqs. ~1!and ~2! are determined by the structure of the scat-ering problem under study; the BSD’s depend on theescription of the illuminating beam.BSD’s are the quantities on which we focus our

ttention in this paper. When the incident longitu-inal electric Ez

i and magnetic Hzi field components

are known, the BSD’s can be determined according to

Îm,TM~g!, ~1 2 g2!Jm~RÎ1 2 g2!exp~igZ!&

51

2p~2i!m *0

2p Ezi

E0exp~2imw!dw, (4)

Fig. 1. Geometry under study.

it

w

wn

Îm,TE~g!, ~1 2 g2!Jm~RÎ1 2 g2!exp~igZ!&

51

2p~2i!m *0

2p Hzi

H0exp~2imw!dw. (5)

B. Plane-Wave Spectrum Approach

The relationship between the distributive approachand the plane-wave spectrum approach relies on theidentity

Im,X~g! 5 *21

11

Im,X~g9!d~g 2 g9!dg9, (6)

n which X stands for TM or TE, d~g 2 g9! designateshe Dirac distribution, and Im,X~g9! can be viewed as

a weight function. The BSD’s are then found to begiven by

UTMi 5

E0

k2 (m52`

1`

~2i!m exp~imw!

3 *21

11

Im,TM~g!Jm~RÎ1 2 g2!exp~igZ!dg, (7)

UTEi 5

H0

k2 (m52`

1`

~2i!m exp~imw!

3 *21

11

Im,TE~g!Jm~RÎ1 2 g2!exp~igZ!dg, (8)

and the weight functions Im,TM~g! and Im,TE~g!, whichcan still be called BSD’s, are given by

Im,TM~g! 5im

4p2~1 2 g2!Jm~RÎ1 2 g2! *0

2p

3 exp~2imw!dw *2`

1` Ezi

E0exp~2igZ!dZ, (9)

Im,TE~g! 5im

4p2~1 2 g2!Jm~RÎ1 2 g2! *0

2p

3 exp~2imw!dw *2`

1` Hzi

H0exp~2igZ!dZ.

(10)

3. Cylindrical Localized Approximation

A. Case under Study

We consider the case of an infinite cylinder arbi-trarily located and arbitrarily oriented in a Gaussianbeam, such as that studied in Ref. 20 within thedistributive approach.

We first consider a Cartesian coordinate systemOGuvw attached to the Gaussian beam. The beam-

aist center of the beam is located at the origin OG ofthis coordinate system. The beam propagates alongthe w axis from negative to positive w ~Fig. 2!. The

wave’s time dependence is chosen to be of the formexp~ivt!, where v is the angular frequency. Thistime-dependence term is omitted in all subsequentequations, as is the normal practice. We assumethat the beam description in terms of electromagneticfields is known with respect to the coordinate systemOGuvw.

The beam description is then converted from thecoordinate system OGuvw into a second coordinatesystem OPu9v9w9 parallel to the coordinate systemOGuvw, which leads to the so-called off-axis Carte-sian description. Coordinates of OG with respect toOP are ~u90, v90, w90!. OP is located on an axis of theinfinite cylinder. This second coordinate systemtakes care of an arbitrary location of the scatterer.

Last, the off-axis Cartesian description is againconverted into a new coordinate system, OPxyz,

hich is rotated with respect to the off-axis coordi-ate system OPu9v9w9 to take care also of arbitrary

orientation of the cylinder ~Fig. 3!. The unit vector eithat defines the direction of propagation of the beam~Fig. 1! is located in the plane ~xOPz!. The z axis isoriented in such a way that the projection of the unit

Fig. 2. Cartesian coordinate systems for beam description.

Fig. 3. Coordinate system for a rotated off-axis description.


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t

2

vector ei onto axis z is negative. Similarly, the x axiss oriented in such a way that the projection of thenit vector ei onto the x axis is negative too. We

define an angle G, reading as

G 5 ~OP zW, 2ei!. (11)

With unit vectors ex and ez on ~OPx! and ~OPz!, re-spectively, we then have

ei 5 2sin Gex 2 cos Gez. (12)

The rotation between OPxyz and OGu9v9w9 is com-pletely defined if we furthermore specify the directionof the electric component Eu9. The general case canbe divided into two cases: parallel, with Eu9 locatedin the plane xOPz, and perpendicular, with Eu9 lo-ated perpendicularly to the plane xOPz. There is a

strong similarity between the two cases,20 so we con-sider only one of them, namely, the perpendicularcase, in this paper. However, the localized approx-imation that we derive can be shown to be valid forboth cases.

Finally, we consider a cylindrical coordinate sys-tem ~z, r, w! attached to the Cartesian system OPxyz~Fig. 1!. The axis of the infinite cylinder is related tohe z axis of the cylindrical coordinate system. How-

ever, because in this paper we deal only with thedescription of the illuminating beam, we do not needto refer explicitly to the properties of the scatterer.

If the illuminating beam is described by a first-order Davis beam,3,4 we obtain the following expres-ions for the longitudinal electric Ez

i and magnetic Hzi

field components ~which are the only ones that werequire for evaluating the BSD’s; see Section 2!:

Ezi 5 2E0 s2CC0CE Q~R sin w 2 U0!, (13)

Hzi 5 H0C0CE@2S 1 2Qs2C~RC cos w 2 ZS 2 V0!#,

(14)

in which

CE 5 exp~iW0!exp@i~RS cos w 1 ZC!#, (15)

C0 5 iQ exp$2iQs2@~RC cos w 2 ZS 2 V0!2

1 ~R sin w 2 U0!2#%, (16)

Q 51

i 2 2s2~RS cos w 1 ZC 1 W0!, (17)

U0, V0, W0! 5 k~u90, v90, w90!, (18)

S 5 sin G, (19)

C 5 cos G. (20)

lso, s is the beam confinement parameter, given by

s 51

kw0, (21)

in which w0 is the beam-waist radius.Equations ~13! and ~14!, however, do not satisfy


Maxwell’s equations ~we say that the description isnot Maxwellian!. A Maxwellian description is ob-tained if we Taylor expand with respect to s and limiturselves to terms up to and including O~s2!, leading

to

Ezi 5 22E0 iC exp~iW0!exp~iZC!~R sin w 2 U0!s

2

3 exp~iRS cos w!, (22)

Hzi 5 H0 exp~iW0!exp~iZC!exp~iRS cos w!~a0 1 a2 s2!,

(23)

in which

a0 5 2S, (24)

a2 5 b0 1 b1 Z 1 b2 Z2, (25)

b0 5 SR2C2 cos2 w 1 ~2iRS2 2 2SRCV0 2 2iRC2!cos w

1 S~U02 1 V0

2! 1 2iCV0 1 2iSW0

1 SR2 sin2 w 2 2RS sin wU0, (26)

b1 5 2S2V0 1 4iCS 2 2RCS2 cos w, (27)

b2 5 S3. (28)

C. Localization Operator

In the case of a cylinder perpendicularly illuminatedby a Gaussian beam, with the incident electric fieldpolarized perpendicularly to the plane defined by thecylinder axis and the incident beam axis at the waist,and the beam-waist center located on the axis of thecylinder, the BSD’s can be approximated by16

Im,TM 5~21!m

2p~1 2 g2! *2`

1` Ez0

i

E0exp~2igZ!dZ, (29)

Im,TE 5~21!m

2p~1 2 g2! *2`

1` Hz0

i

H0exp~2igZ!dZ, (30)

defining what we call the cylindrical localized approx-imation, in analogy with the case of the GLMT forspheres.12–15 In these equations we have introducedEz0

i and Hz0

i according to

~Ezi , Hz

i! 5 ~Ez0

i , Hz0

i !exp~iR cos w!, (31)

in which exp~iR cos w! is the plane-wave contributionor perpendicular illumination ~S 5 1! and an overbar

designates a quantity evaluated within the frame-work of the localized approximation. For Ez0

i and Hz0

i

this means that

~Ez0

i , Hz0

i ! 5 G~Ez0

i , Hz0

i !, (32)

in which G is the so-called localization operator thatreplaces R with m and w with py2 in any expressiono which it is applied:

G:R3m, w3 py2. (33)

I

st

drw

N

0

t

t

i

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oE

To establish a localized approximation in the gen-eral case studied in this paper we tentatively gener-alize Eqs. ~29! and ~30! to

Im,TM~g! 5K

~1 2 g2! *2`

1` Ez0

i

E0exp~2igZ!dZ, (34)

Im,TE~g! 5K

~1 2 g2! *2`

1` Hz0

i

H0exp~2igZ!dZ, (35)

in which K is a prefactor to be evaluated. Also, Eq.~31! now reads as

~Ezi , Hz

i! 5 ~Ez0

i , Hz0

i !exp~iRS cos w!, (36)

and the localization operator G is generalized to

G:R3R0, w3 w0. (37)

If we find a way to determine the localization op-erator ~i.e., to determine R0 and w0! so Im,TM~g! andIm,TE~g! are good approximations of the exact BSD’sm,TM~g! and Im,TE~g!, then we have simultaneously

established a cylindrical localized approximation andjustified it.

C. Justification

The localized approximation for cylinders ~as forpheres12–15! is based on the first-order Davis descrip-ion.3 It is convenient first to examine the TE BSD

@Eq. ~35!#, so we consider the expression Hzi given by

Eq. ~23!, from which we extract

Hz0

i 5 H0 exp~iW0!exp~iZC!~a0 1 a2 s2!. (38)

The localized approximation must hold indepen-ently for each term that has a given power withespect to the beam confinement parameter. Then,orking out the O~s0! term, Eq. ~35! leads to

Im,TE0 ~g! 5

K~1 2 g2! *

2`

1` Hz0us0i

H0exp~2igZ!dZ, (39)

in which

Hz0us0i

H05

Hz0us0i

H05 2S exp~iW0!exp~iZC!. (40)

ote that, at O~s0!, R and w are not involved in Hz0

i ,leading to the fact that Hz0us0

i 5 Hz0us0i , irrespective of

the localized operator. Inserting Eq. ~40! into Eq.~39! and taking advantage of the following represen-tation of the Dirac distribution:

*2`

1`

exp@iZ~x 2 x9!#dZ 5 2pd~x9 2 x!, (41)

we obtain

Im,TE0 ~g! 5 22pKS exp~iW0!

d~g 2 C!

1 2 g2 . (42)

But the exact expression for the O~s ! contribution tohe TE BSD reads as20

Im,TE0 ~g! 5 2

~21m!

Sexp~iW0!d~g 2 C!. (43)

However, we have the following identity between dis-tributions ~Appendix A!:

d~g 2 C!

1 2 g2 5d~g 2 C!

1 2 C2 5d~g 2 C!

S2 , (44)

leading to

K 5~21!m

2p, (45)

which establishes that the prefactor K in our generalcase @Eqs. ~34! and ~35!# is the same as in our specialcase of perpendicular illumination @Eqs. ~29! and~30!#. This result was indeed expected.

Next, from Eqs. ~35!, ~45!, and ~38!, the O~s2! con-ribution to the TE BSD reads as

Im,TE2 ~g! 5

~21!m

2p~1 2 g2!exp~iW0!

3 Hb0 *2`

1`

exp@iZ~C 2 g!#dZ

1 b1 *2`

1`

exp@iZ~C 2 g!#ZdZ

1 b2 *2`

1`

exp@iZ~C 2 g!#Z2dZJ , (46)

n which we set

Im,TE~g! 5 Im,TE0 ~g! 1 s2Im,TE

2 ~g! (47)

nd bi is G~bi!, i.e., bi to which we apply the localiza-tion operator G. With the representation @Eq. ~41!#f the Dirac distribution and its first two derivatives,q. ~47! becomes

Im,TE2 ~g! 5 ~21!m exp~iW0!Fb0

d~g 2 C!

1 2 g2

1 ib1

d9~g 2 C!

1 2 g2 2 b2

d0~g 2 C!

1 2 g2 G . (48)

We then use Eq. ~44! again and also ~Appendix A!

d9~g 2 C!

1 2 g2 5d9~g 2 C!

S2 22CS4 d~g 2 C!, (49)

d0~g 2 C!

1 2 g2 5d0~g 2 C!

S2 24CS4 d9~g 2 C!

12S4 S1 1

4C2

S2 Dd~g 2 C! (50)


r

tt

p@to

E

oedi

rwch

2

to obtain

Im,TE2 ~g! 5 ~21!m exp~iW0!FSb0

S2 2 ib1

2CS4 2 2

b2

S4

2 8C2

S6 b2Dd~g 2 C! 1 Sib1

S2 14CS4 b2Dd9~g

2 C! 2b2

S2 d0~g 2 C!G . (51)

Equation ~51! is to be compared with the exact ex-pression of the O~s2! contribution to the TE BSDeading as20

Im,TE2 ~g! 5 (

i50

2

Im,TE2i d~i!~g 2 C!, (52)

in which

Im,TE20 5

~21!m

S2 exp~iW0!FS~U02 1 V0

2! 1 2iSW0

2 2iCV0 2 2S 1 2mU0 1m2

S G , (53)

Im,TE21 5 2i~21!mV0 exp~iW0!, (54)

Im,TE22 5 2~21!mS exp~iW0!. (55)

We then compare Eqs. ~51! and ~52!. With b2 5 S3

@Eq. ~28!#, we have G~b2! 5 b2 5 S3 too, and it is foundthat the terms that involve d0~g 2 C! are identical andherefore provide no information. For the termshat involve d9~g 2 C! we still have to use b2 5 S3 but

also, from Eq. ~27!,

b1 5 2S2V0 1 4iCS 2 2R0 CS2 cos w0. (56)

It is then found that the following relation must besatisfied:

2i~V0 2 R0 cos w0! 5 2iV0, (57)

leading to the prescription

w0 5 py2, (58)

which is again the same prescription as in our specialcase of perpendicular illumination.

Finally, using this prescription, we find that thecomparison between the d~g 2 C! terms in Eqs. ~51!and ~52! leads to

1S

~U02 1 V0

2! 1 2iW0

S2 2iC

V0

S2 22S

2 2R0

U0

S1

R02

S

51S

~U02 1 V0

2! 1 2iW0

S2 2iC

V0

S2 22S

1 2mU0

S2 1m2

S3 ,

(59)

leading to the prescription

R0 5 2mS

. (60)


This prescription is different from the one for per-endicular illumination, for which we had R0 5 mrelations ~33!#, but is compatible with it because, inhe special case, we have S 5 1 and, furthermore,nly even powers of R0 appeared in the formulation,

so the minus in Eq. ~60! is irrelevant.We now check how the localized approximation es-

tablished above behaves for the TM BSD. InsertingEqs. ~58! and ~60! into Eq. ~22!, we first establish that

Ez0

i

E05 22iC exp~iW0!exp~iZC!S2

mS

2 U0Ds2. (61)

This result is inserted into Eq. ~34!, with K given byq. ~45!, leading to

Im,TM~g! 5 2i~21!m CS2 SU0 1

mSDexp~iW0!s

2d~g 2 C!,

(62)

which agrees perfectly with the exact expression ofthe TM BSD Im,TM~g! given in Ref. 20.

In summary, the localized approximation is givenby Eqs. ~29! and ~30!, with the localization operatorgiven by

G:R32mS

, w3 py2. (63)

The approximate BSD’s then agree perfectly withthe exact ones obtained from the Maxwellian contri-bution to the first-order Davis beam description.

4. Anticipation of Higher-Order Descriptions

After the first-order Davis beam description we canuse higher-order Davis beam descriptions, preferablyin their symmetrized versions, which provide refineddescriptions of Gaussian beams.4,21,22 In particular,a fifth-order Davis beam then provides Maxwelliancontributions up to terms that are O~s10!. It hasbeen found for in the case of the GLMT for spheresthat, although it is based on a first-order Davis beam,the ~spherical! localized approximation correctly an-ticipates the behavior of higher-order descriptions,leading to the introduction of so-called standardbeams.14,15,22 In this section we examine whetherthis property is also shared by the cylindrical local-ized approximation. We do this by comparing theexpressions for the BSD’s Im,TM~g! and Im,TE~g!, basedn a first-order Davis beam description, with exactxpressions evaluated for a fifth-order Davis beamescription. These comparisons have to be done bynclusion of terms up to O~s10!, because a fifth-order

Davis beam is Maxwellian up to O~s10! terms. Theequired algebra is extensive and was carried outith a symbolic computation software ~Maple!. Ac-

ordingly, the resultant expressions are not givenere in extenso.

i

U

B

g

A ~ j!
We start with the approximate Im,TE~g! given byEq. ~30! in which we now have, from Eq. ~14!,
A 5Hz0

i

H05 2i exp~iW0!exp~iZC!Q#

3 expF2iQ# s2SZ2S2 1 V02 1 2V0 ZS 1

m2

S2

1 U02 1 2m

U0

S DG@S 1 2Q# s2C~ZS 1 V0!#, (64)

in which

Q# 51

i 2 2s2~ZC 1 W0!. (65)

In Eq. ~64! we retain only terms that include O~s10! byperforming a Taylor expansion, leading to

A 5 exp~iW0!exp~iZC! (j50

5

b2js2j, (66)

n which the coefficients b2j take the form

b2j 5 (i50

2j

ai2jZi. (67)

Inserting Eq. ~66! into Eq. ~30!, we obtain

Im,TE~g! 5~21!m exp~iW0!

2p~1 2 g2! *2`

1` S(j50

5

b2js2jD

3 exp@iZ~C 2 g!#dZ. (68)

sing Eq. ~67!, we may write Eq. ~68! as

Im,TE~g! 5~21!m exp~iW0!

2p~1 2 g2! (j50

10

Aj *2`

1`

Zj

3 exp@iZ~C 2 g!#dZ, (69)

in which

Aj 5 (l5int~ j11!y2

5

aj2ls2l. (70)

ut, from Eq. ~41!, we derive

*2`

1`

Zj exp@iZ~C 2 g!#dZ 5 2pijd~ j!~g 2 C!, (71)

leading to

Im,TE~g! 5 ~21!m exp~iW0! (j50

10

Ajij d~ j!~g 2 C!

1 2 g2 . (72)

We can then use Eqs. ~44!, ~49!, and ~50!, and, moreenerally, a procedure similar to the one given in

ppendix A to evaluate all the distributions d ~g 2C!y~1 2 g2! involved in Eq. ~72! in the form

d~ j!~g 2 C!

1 2 g2 5 (l50

j

~21!lCjld~l !~g 2 C!. (73)

Inserting Eq. ~73! into Eq. ~72!, we find that Im,TE~g!takes the form

Im,TE~g! 5 2~21!m exp~iW0! (j50

5

a2js2j, (74)

which possesses the more general form

Im,TE~g! 5 (j50

5

Im,TE2j ~g!s2j. (75)

The evaluation of a0 and a2 then allows us to eval-uate Im,TE

0 ~g! and Im,TE2 ~g!, which we find agree ex-

actly with Eqs. ~43! and ~52!, respectively. This re-sult confirms the validity of the localizedapproximation, based on the first-order Davis beamdescription containing terms that are O~s0! and O~s2!.

The coefficients a4, . . . , a10, however, contain thenew information that we are looking for, concerninghigher-order Davis beams. These coefficients areactually rather lengthy, but fortunately we do notneed them in extenso. Indeed, it will be sufficient toconsider subterms of each a2j as follows:

Each coefficient a2j is actually found to be a distri-bution that has the general form

a2j 5 (l50

2j

a2jld~l !~g 2 C!. (76)

Each coefficient a2jl takes the form of an expansion

in powers of m. Retaining in each of these coeffi-cients the term that exhibits the highest power withrespect to m, we define the leading part L~a2j! of thedistribution a2j. These leading parts read as

L~a4! 51

2S5 m4d~g 2 C! 12S2 SiV0 2

CSDm2d9~g 2 C!

2m2

Sd0~g 2 C! 1 2S~C 2 iSV0!d

~3!~g 2 C!

112

S3d~4!~g 2 C!, (77)

L~a6! 5 21

6S7 m6d~g 2 C! 11S4 S2C

S2 iV0Dm4d9~g 2 C!

11

2S3 m4d0~g 2 C! 1 2SiV0 2 2CSDm2d~3!~g 2 C!

212

Sm2d~4!~g 2 C! 1 S3~2C 2 iSV0!d~5!~g 2 C!

116

S5d~6!~g 2 C!, (78)


a

2

L~a8! 51

24S9 m8d~g 2 C! 11S6 S13 iV0 2

CSDm6d9~g 2 C!

21

6S5 m6d0~g 2 C! 11S2 S3 C

S2 iV0Dm4d~3!~g

2 C! 11

4Sm4d~4!~g 2 C!

1 S~iSV0 2 3C!m2d~5!~g 2 C!

216

S3m2d~6!~g 2 C!

1 S5SC 213

iSV0Dd~7!~g 2 C! 1124

S7d~8!~g 2 C!,

(79)

L~a10! 5 21

120S11 m10d~g 2 C!

11

3S8 SCS 214

iV0Dm8d9~g 2 C!

11

24S7 m8d0~g 2 C!

11

3S4 SiV0 2 4CSDm6d~3!~g 2 C!

21

12S3 m6d~4!~g 2 C!

1 S2 CS

212

iV0Dm4d~5!~g 2 C!

1112

Sm4d~6!~g 2 C!

1S3

3~iSV0 2 4C!m2d~7!~g 2 C!

2124

S5m2d~8!~g 2 C!

113

S7SC 214

iSV0Dd~9!~g 2 C!

11

120S9d~10!~g 2 C!. (80)

These results are to be compared with the corre-sponding results for the exact TE BSD, Im,TE~g!.However, in this section Im,TE~g! must be evaluatednot on a first-order Davis beam but on a fifth-orderDavis beam, which is Maxwellian up to terms thatinclude O~s10!. This evaluation was carried out pre-viously,23 although the resultant expressions weretoo lengthy to be published in extenso. Instead, they


were saved in Maple files. From these files, Im,TE~g!takes a form similar to Eq. ~74!:

Im,TE~g! 5 2~21!m exp~iW0! (j50

5

a2js2j. (81)

Here a0 and a2 again agree perfectly with a0 and2, as expected. Similarly as for a2j, we then define

the leading part L~a2j! of a2j. The leading parts ofthe distributions a4, . . . , a10 read as

L~a4! 51

2S5 m4d~g 2 C! 12S2 SiV0 1

CS

pDm2d9~g 2 C!

2m2

Sd0~g 2 C! 1 2S~C 2 iSV0!d

~3!~g 2 C!

112

S3d~4!~g 2 C!, (82)

L~a6! 5 21

6S7 m6d~g 2 C!

11S4 S2iV0 2 2

CS

pDm4d9~g 2 C!

11

2S3 m4d0~g 2 C! 1 2~iV0 1 0*!m2d~3!~g 2 C!

212

Sm2d~4!~g 2 C! 1 S3~2C 2 iSV0!d~5!~g 2 C!

116

S5d~6!~g 2 C!, (83)

L~a8! 51

24S9 m8d~g 2 C!

11S6 S13 iV0 1

CS

pDm6d9~g 2 C!

21

6S5 m6d0~g 2 C!

11S2 S2 C

Sp 2 iV0Dm4d~3!~g 2 C!

11

4Sm4d~4!~g 2 C! 1 S~iSV0

2 C*!m2d~5!~g 2 C! 216

S3m2d~6!~g 2 C!

1 S5SC 213

iSV0Dd~7!~g 2 C! 1124

S7d~8!~g 2 C!,

(84)

mtfae

tpW

f

L~a10! 5 21

120S11 m10d~g 2 C!

11

3S8 S2 CS

p 214

iV0Dm8d9~g 2 C!

11

24S7 m8d0~g 2 C!

11

3S4 SiV0 1 2CS

pDm6d~3!~g

2 C! 21

12S3 m6d~4!~g 2 C!

1 S0* 212

iV0Dm4d~5!~g 2 C!

1112

Sm4d~6!~g 2 C!

1S3

3~iSV0 2 2C*!m2d~7!~g 2 C!

2124

S5m2d~8!~g 2 C!

113

S7SC 214

iSV0Dd~9!~g 2 C!

11

120S9d~10!~g 2 C!. (85)

The agreement between L~a2j! and L~a2j! is re-arkable, except for the few terms marked with as-

erisks. These terms are all exactly 0 for C 5 0, i.e.,or perpendicular illumination ~not necessarily onxis; i.e., the axis of the Gaussian beam may notncounter the axis of the cylinder!. When C in-

creases from 0, the error terms become more impor-tant, but, in any case, they make only a partialcontribution to the scattering phenomena that, asexpected, is small. This statement is verified nu-merically below. In particular, we can also definethe leading parts L@L~a2j!# and L@L~a2j!# of the lead-ing parts L~a2j! and L~a2j!, respectively, as the oneshat correspond to the terms that involve the highestower of m, whatever the associated distribution.e then readily find that

L@L~a2j!# 5 L@L~a2j!#. (86)

The same results can be achieved for the TM dis-tributions. The procedure is similar to the oneabove for the TE distributions, with Eq. ~29! insteadof Eq. ~30!, and with Ez0

i yE0 evaluated for a first-order

Davis beam. Then, instead of using Eq. ~14! leadingto Eq. ~64!, we now use Eq. ~13!, which leads to

Ez0

i

E05 22is2C exp~iW0!exp~iZC!Q# 2

3 expF2iQ# s2SZ2S2 1 V02 1 2V0 ZS

1m2

S2 1 U02 1 2m

U0

S DGSmS

1 U0D . (87)

Instead of Eq. ~74!, we then find that

Im,TM~g! 5 2~21!m exp~iW0! (j51

5

c2js2j, (88)

in which the summation starts from j 5 1 instead ofrom j 5 0. The coefficient c2 is then found to con-

tribute to Im,TM~g! of Eq. ~88!, which identifies exactlywith Im,TM~g! of Eq. ~62!. Again, this result confirmsonce more the validity of the localized approximation,based on the first-order Davis beam description.

The leading terms of the other coefficients arefound to be

L~c4! 5 2iCS5 m3d~g 2 C! 2 4

CS2 V0 md9~g 2 C!

2 2iCS

md0~g 2 C!, (89)

L~c6! 5 2iCS7 m5d~g 2 C!

1 4CS4 Si

CS

1 V0Dm3d9~g 2 C!

1 2iCS3 m3d0~g 2 C!

2 4CSiCS

1 V0Dmd~3!~g 2 C!

2 iCSmd~4!~g 2 C!, (90)

L~c8! 513

iCS9 m7d~g 2 C!

2 2CS6 SV0 1 2i

CSDm5d9~g 2 C!

2 iCS5 m5d0~g 2 C!

14CS2 SV0 1 2i

CSDm3d~3!~g 2 C!

1 iCS

m3d~4!~g 2 C! 2 2CS~2iC

1 SV0!md~5!~g 2 C!

213

iCS3md~6!~g 2 C!, (91)


l

a

2

L~c10! 5 2112

iC

S11 m9d~g 2 C!

1 2CS8 S1

3V0 1 i

CSDm7d9~g 2 C!

113

iCS7 m7d0~g 2 C!

22CS4 S3i

CS

1 V0Dm5d~3!~g 2 C!

212

iCS3 m5d~4!~g 2 C!

1 2CS3iCS

1 V0Dm3d~5!~g 2 C!

113

iCSm3d~6!~g 2 C!

2 2CS3S13

SV0 1 iCDmd~7!~g 2 C!

2112

iCS5md~8!~g 2 C!. (92)

The exact expression for Im,TM~g!, relying on a fifth-order Davis description, can again be found from thestudy described in Ref. 23. It reads as

Im,TM~g! 5 2~21!m exp~iW0! (j51

5

c2js2j. (93)

We then find that c2 5 c2, in agreement again withour previous justification of the localized approxima-tion. For the other distributions c2j, we have thefollowing leading expressions:

L~c4! 5 2iCS5 m3d~g 2 C! 1 F24

CS2 V0

12iS S1 1

2C2

S2 D*G3 md9~g 2 C! 2 2i

CS

md0~g 2 C!, (94)

L~c6! 5 2iCS7 m5d~g 2 C! 1

4CS4 FV0

11

2C S3iS 24iSD*G

3 m3d9~g 2 C! 1 2iCS3 m3d0~g 2 C!

2 4CSV0 2iS2C

*D3 md~3!~g 2 C! 2 iCSmd~4!~g 2 C!, (95)


L~c8! 513

iCS9 m7d~g 2 C! 2

2CS6 FV0

2 iS S2C

1 3CSD*G

3 m5d9~g 2 C! 2 iCS5 m5d0~g 2 C!

14CS2 FV0 2 iSC

S1

S2CD*Gm3d~3!~g 2 C!

1 iCS

m3d~4!~g 2 C!

2 2CSFSV0 1 iSC 212

S2

CD*G3 md~5!~g 2 C! 2

13

iCS3md~6!~g 2 C!, (96)

L~c10! 5 21

12i

CS11 m9d~g 2 C! 1

2CS8 F1

3V0

213

iS12

SC

1 4CSD*G 3 m7d9~g 2 C!

113

iCS7 m7d0~g 2 C!

2 2CS4 FV0 2 iS1

2SC

1 2CSD*G

3 m5d~3!~g 2 C! 212

iCS3 m5d~4!~g 2 C!

1 2CSV0 2 iS

2C*Dm3d~5!~g 2 C! 1

13

iCSm3d~6!

3 ~g 2 C!

2 2CS3F13

SV0 113

iS2C 212

S2

CD*G3 md~7!~g 2 C! 2

112

iCS5md~8!~g 2 C!. (97)

The comparison of L~c2j! and L~c2j! then deservesthe same comments as for the comparison of L~a2j!and L~a2j!. In particular, the leading parts of theeading parts are identical:

L@L~c2j!# 5 L@L~c2j!#. (98)

We therefore found that the localized approxima-tion, although it is based on a first-order Davis beam,anticipates the behavior of higher-order beam de-scriptions. This result was in fact expected becausesuch behavior was also observed in the case of thelocalized approximation in spherical coordinates.

5. Algebraic Expressions

When we compare Eqs. ~29! and ~30! for the localizedpproximation and nonapproximated expressions ~9!

tm

3i

a

wbi

rwo

t

and ~10!, it is obvious that the localized approxima-tion will allow us to speed up the numerical calcula-tions. A more effective speedup may beaccomplished if Eqs. ~29! and ~30! are amenable toalgebraic expressions. This is the issue consideredin this section. We start by considering the specialcase of perpendicular illumination, for which twomethods will be used.

A. Perpendicular Illumination: First Method

The special case of perpendicular illumination is de-fined by

C 5 0, S 5 1. (99)

For on-axis perpendicular illumination, the axis ofhe Gaussian beam meets the axis of the cylinder. Aore restricted case occurs when U0 5 V0 5 W0 5 0.

This was the case previously discussed ~Subsection.B and Ref. 16!. Here we only assume Eqs. ~99!;.e., we deal with off-axis perpendicular illumination.

For the TE BSD, we use Eq. ~30! with Hz0

i yH0 sym-plifying to @from Eqs. ~64!, ~65! and ~99!#

Hz0

i

H05 2iQ# exp~iW0!exp@2iQ# s2~Z2 1 2V0 Z 1 V0

2

1 m2 1 U02 1 2mU0!#, (100)

in which

Q# 51

i 2 2s2W0, (101)

leading to

Im,TE~g! 52i~21!m

2p~1 2 g2!

exp~iW0!

i 2 2s2W0exp~2aT!I, (102)

in which

a 5s2 2 2is4W0

1 1 4s4W02 , (103)

T 5 V02 1 m2 1 U0

2 1 2mU0, (104)

and I is an integral that reads as

I 5 *2`

1`

exp~2aZ2!exp~ibZ!dZ, (105)

in which

b 5 2iaV0 2 g. (106)

But ~from Appendix B!

*2`

1`

exp~2aZ2!exp~ibZ!dZ

5 Îp

aexpS2b2

4a D , Re~a! . 0, (107)

leading to the final algebraic result for perpendicularillumination:

Im,TE~g! 52i~21!m

2p~1 2 g2! Îp

aexp~iW0!

i 2 2s2W0

3 exp~2aT!expS2b2

4a D , (108)

in which a, T, and b are given by Eqs. ~103!, ~104!,nd ~106!, respectively.Let us note that in Ref. 16 we used

*0

1`

exp~2ax2!cos~bx!dx

512 Îp

aexpS2b2

4a D , a . 0. (109)

Equation ~107! is a generalization of Eq. ~109! inhich the quantity a is allowed to be a complex num-er. This generalization is valid, as we demonstraten Appendix B.

B. Perpendicular Illumination: Second Method

The second method relies on Eq. ~109!, assuming thatwe are not aware of the validity of Eq. ~107!. Theesult will take the form of a series to be truncated,hich is less appealing than our result @Eq. ~108!# butpens the way to the most general case.To take advantage of Eq. ~109! we rewrite the in-

egral I in the form

I 5 *2`

1`

exp~iaZ2!exp~2bZ!exp~2mZ2!exp~inZ!dZ,

(110)

in which a, b, m, and n pertain to 5 according to

a 5 2Im~a!, (111)

b 5 Im~b!, (112)

m 5 Re~a!, (113)

n 5 Re~b!. (114)

We then expand exp~iaZ2! and exp~2bZ!, leadingto

I 5 (n50

`

(m50

` ~ia!n

n!~2b!m

m!Jnm, (115)

in which Jnm are integrals that read as

Jnm 5 *2`

1`

Z2n1m exp~2mZ2!exp~inZ!dZ. (116)


Tt

w

t

dwd1

2

Next, from Eq. ~109!, we can establish that

*2`

1`

exp~2mZ2!exp~inZ!dZ

5 Îp

mexpS2n2

4m D , m . 0, n [ 5, (117)

which can be derived to provide

*2`

1`

Zj exp~2mZ2!exp~inZ!dZ

51ij Îp

m

] j

]n j expS2n2

4m D . (118)

Inserting this result into Eq. ~115!, we then have

I 5 Îp

m (n50

` ~2ia!n

n!]2n

]n2n (m50

` ~ib!m

m!]m

]nm expS2n2

4m D , (119)

allowing us to evaluate Im,TE~g! to arbitrary accuracy,in principle. In practice, however, we have to dealwith truncations with respect to powers of s. Tomake these powers more apparent, we set

F 5 1 1 4s4W02, (120)

so we have

a 52W0 s4

F, (121)

b 52V0 s2

F. (122)

In these forms, a ; s4 and b ; s2, so it is advanta-geous to reverse the order of summations in Eq. ~119!to evaluate truncations up to a certain order O~sn!.

herefore, truncations are best evaluated accordingo

I 5 Îp

m (m50

` S2iV0 s2

F Dm 1m!

]m

]nm

3 (n50

` S22iW0 s4

F Dn 1n!

]2n

]n2n expS2n2

4m D . (123)

In practice, a truncation up to terms that includesO~s10! ~where s appears explicitly! is likely to be suf-ficient for many applications. To appreciate thisstatement, recall that such a truncation would corre-spond to a fifth-order Davis beam description. Wethen obtain

I 5 Îp

mexpS2n2

4m D (j50

5

i2js2j, (124)

in which

i0 5 1, (125)


i2 5 2iV0

Fmn, (126)

i4 51

Fm SV02

F1 iW0DS1 2

n2

2mD , (127)

i6 5V0n

F2m2 S3W0 2 iV0

2

F DS1 2n2

6mD , (128)

i8 51

F2m2 S V04

2F2 1 3iV0

2W0

F2

32

W02D

3 S1 2n2

m1

112

n4

m2D , (129)

i10 5n

F3m3 S152

iV0 W02 1 5

V03W0

F2

12

iV0

5

F2D3 S1 2

13

n2

m1

160

n4

m2D . (130)

Obviously, the two methods are equivalent, as wecan be show from Eq. ~119! by establishing first that

(m50

` ~ib!m

m!]m

]nm expS2n2

4m D5 (

m50

` ~ib!m

m!]m

]~n 1 ib!m expF2~n 1 ib!2

4m GUib50

5 expF2~n 1 ib!2

4m G (131)

and next that

I 5 Îp

m (n50

` ~2ia!n

n!]2n

]n2n expF2~n 1 ib!2

4m G5 Î p

m 2 iaexpF2~n 1 ib!2

4~m 2 ia! G , (132)

which, when we use Eqs. ~111!–~114!, is identifiedith Eq. ~107!.For the TM BSD, Im,TM~g!, because Ez

i is propor-ional to C @Eq. ~13!# we immediately have

Im,TM~g! 5 0. (133)

C. General Case

For the general case we use the truncation method ofSubsection 5.B. Im,TE~g! is again given by Eq. ~30!and Hz0

i yH0 is given by Eqs. ~64! and ~65!. The newifficulty here is that the rescaled coordinate Z, onhich we integrate in Eq. ~30!, is involved in theenominator of Q# . We then use expansions ofy~1 1 x! and of ~a 1 b!n and

(k51

`

(l50

k21

5 (l50

`

(k5l11

`

(134)

a

Fpsiti

a

to establish that

Q# 5 2i (l50

`

Al ZlCl, (135)

in which

Al 5 (k5l11

`

~22is2!k21Sk 2 1l DW0

k212l

5 ~22is2!l~1 1 2is2W0!2l21. (136)

We then obtain

Q# 5 (l50

`

ql Zl, (137)

in which

ql 5 2i~22is2C!l

~1 1 2is2W0!l11 , (138)

so Im,TE~g! is found to read as

Im,TE~g! 52i~21!m

2p~1 2 g2!exp~iW0!

3 @SJ1 1 2s2C~SJ2 1 V0 J3!#, (139)

in which J1, J2, J3 are integrals that read as

J1 5 *2`

1` S(l50

`

ql ZlDexpF2is2S(

l50

`

ql ZlD

3 ~Z2S2 1 2V0 ZS 1 T!Gexp@iZ~C 2 g!#dZ,

(140)

J2 5 *2`

1` S(l50

`

ql ZlD2

Z expF2is2S(l50

`

ql ZlD

3 ~Z2S2 1 2V0 ZS 1 T!Gexp@iZ~C 2 g!#dZ,

(141)

J3 5 *2`

1` S(l50

`

ql ZlD2

expF2is2S(l50

`

ql ZlD 3 ~Z2S2

1 2V0 ZS 1 T!Gexp@iZ~C 2 g!#dZ (142)

nd T generalizes Eq. ~104! to

T 5 V02 1

m2

S2 1 U02 1 2m

U0

S. (143)

Next, we show that

S(`

ql ZlD2

5 (`

ql3Zl, (144)

l50 l50

in which

ql3 5 (

i50

l

qiql2i, (145)

and

S(l50

`

ql ZlD2

Z 5 (l50

`

ql2Zl, (146)

in which

q02 5 0 (147)

ql2 5 (

i50

l21

qiql212i, l > 1. (148)

We also establish that

S(l50

`

ql ZlD~Z2S2 1 2V0 ZS 1 T! 5 (

l50

`

El Zl, (149)

in which

El 5 ε1l S2ql22 1 2ε0

l V0 Sql21 1 Tql, (150)

with

ε0l 5 H0 l 5 0

1 otherwise , (151)

ε1l 5 H0 l 5 0, 1

1 otherwise . (152)

Assembling everything, we then obtain

Im,TE~g! 52i~21!m

2p~1 2 g2!exp~iW0! (

l50

`

kl Kl, (153)

in which kl are coefficients that read as

kl 5 Sql 1 2s2C~Sql2 1 V0 ql

3! (154)

and Kl are integrals that read as

Kl 5 *2`

1`

Zl expS2is2 (l50

`

El ZlD 3 exp@iZ~C 2 g!#dZ.

(155)

rom Eqs. ~153!–~155! it is possible to derive an ex-ression similar to Eq. ~119! or ~123!. This expres-ion is, however, rather awkward and is of littlenterest. Rather, we shall be content in deriving aruncated expression of Eq. ~153!, up to terms thatnclude O~s10! in the sense that we rewrite ql as

ql 52i~22is2C!l

Gl11 (156)

nd state that ql is O~s2l!, with

G 5 1 1 2is2W0. (157)


n

i

2

2

An examination of Eq. ~154! then reveals that weneed to evaluate only q0, . . . , q5, which are found tobe

q0 5 21G SiS 1 2s2 CV0

G D , (158)

qj 5 4ajSs2CiG D jq0, j 5 1, . . . , 5, (159)

in which ~a1, . . . , a5! 5 ~1, 3, 8, 20, 48!. Of course,the term that is O~s12! in q5 can be omitted.

We next consider the integral K0:

K0 5 *2`

1`

expS2is2(l50

`

El ZlDexp@iZ~C 2 g!#dZ. (160)

Because the lowest order in q0 is O~s0!, we need toretain up to O~s10! terms in K0. Then the summa-tion ¥l50

` can be truncated to ¥l506 because by exam-

ining Eq. ~150! we find that the lowest-order term inE7 is O~s10!, leading to an O~s12! term in the firstexponential of the integrand of Eq. ~160!. Then, af-ter a bit of algebra, we find that we need to evaluatea contribution to K0 that reads as ~we preserve the

otation K0 for convenience!

K0 5 exp~2is2E0! *2`

1`

exp~s2E1i Z!exp~2is2E2

riZ2!

3 exp~2is2E3 Z3!exp~2is2E4 Z4!exp~2is2E5 Z5!

3 exp~2is2E6 Z6!exp~2mS2Z2!exp~iZu!dZ,(161)

n which

E1i 5 Im~E1!, (162)

E2ri 5 Re~E2! 1 iFIm~E2! 1

S2

F G , (163)

u 5 C 2 g 2 s2 Re~E1!, (164)

where m is given by Eq. ~113!.Equation ~161! can afterward be expanded to

K0 5 exp~2is2E0! (n150

` ~s2E1i !n1

n1!(n250

` ~2is2E2ri!n2

n2!

3 (n350

` ~2is2E3!n3

n3!(n450

` ~2is2E4!n4

n4!(n550

` ~2is2E5!n5

n5!

3 (n650

` ~2is2E6!n6

n6! *2`

1`

Zn112n213n314n415n516n6

3 exp~2mS2Z2!exp~iZu!dZ. (165)


In this integral we have mS . 0 and u [ 5, so wemay use Eqs. ~117! and ~118!. We may then rewriteEq. ~165! as

K0 5 Î p

mS2 exp~2is2E0! (n150

` ~2s2E1i !n1

n1!]n1

]un1

3 (n250

` ~is2E2ri!n2

n2!]2n2

]u2n2 (n350

` ~s2E3!n3

n3!]3n3

]u3n3

3 (n450

` ~2is2E4!n4

n4!]4n4

]u4n4 (n550

` ~2s2E5!n5

n5!]5n5

]u5n5

3 (n650

` ~is2E6!n6

n6!]6n6

]u6n6expS 2u2

4mS2D , (166)

which is best evaluated by use of symbolic computa-tions. Once K0 is evaluated, we can evaluate theother required integrals Kl by using

Kl 5 ~2i!l ]lK0

]ul . (167)

Pruning all the terms whose order is larger thanO~s10!, we finally obtain

Im,TE~g! 52i~21!m

2p~1 2 g2!exp~iW0!Î p

mS2

3 exp~2is2E0!expS 2u2

4mS2D (j50

5

e2js2j. (168)

Coefficients e2j, at this stage, are saved in symbolicfiles. However, we provide some of them here asfollows:

e0 5 2iSG

, (169)

e2 5 21

G2 F2CV0 112

i~4C 1 Ei

1G!

Su

mG , (170)

e4 51

4G3F2mSES1 2

u2

2S2mD 112

iCu

G3m2S2

3 F~2iV0 Ei1mG 1 8iCV0m 1 3S! 2

12

u2

mSG , (171)

in which

E 5 4S2W0 G2F 1 i~Ei1!2G2F2 1 8iCEi

1GF2

1 8CV0 G2S 1 24iC2F2. (172)

i

d

fi

i

r

ti1

ft

f

f

For Im,TM~g! we start from Eq. ~29! with Ez0yE0

given by Eqs. ~87! and ~65! and use the same proce-ure as for the TE BSD, leading to

Im,TM~g! 5 22i~21!m

2p~1 2 g2!s2CSm

S1 U0D

3 exp~iW0!Î p

mS2 exp~2is2E0!

3 exp2u2

4mS2 (j50

4

f2js2j, (173)

where the coefficients f2j are again saved in symbolicles. Among them we have

f0 5 21

G2 (174)

f2 5 212

Ei1G 1 4CG3mS2 u, (175)

f4 52i

4G4S2F2mES1 2

u2

2S2mD1

12

Cu

G4m2S2 S3 212

u2

mS2D , (176)

n which E is given by Eq. ~172!.

6. Numerical Results

The numerical results presented in this section con-cern a discussion of the behavior of the BSD’s pre-dicted in the framework of the localizedapproximation ~Subsection 6.A! and a comparison ofthe incident beam reconstructed by the BSD’s com-puted in the framework of the localized approxima-tion and the incident beam directly computed from afirst-order Davis beam ~Subsection 6.B!.

A. Behavior of Beam-Shape Distributions

From Eqs. ~29! and ~30!, the BSD’s depend on s,which defines the degree of focusing of the beam ~s 50 for a plane wave!; G, the angle between the directionof propagation of the incident beam and the axis ofthe cylinder; and m, the order of the BSD. Figure 4displays Im,TE~g! and Im,TM~g! versus g, with s 51022, G 5 py4, and m 5 0. Our main remarks hereare that Im,TM~g! is nearly zero and that Im,TE~g! isepresented by a peak centered on g 5 cos G.

The series of Figs. 5–7 focuses on the behavior ofhis peak when computational parameters are mod-fied. In Fig. 5 the beam parameter is now s 5 5 3022; the width of the peak is more important,

whereas its height is less than in Fig. 4. In Fig. 6, mis 100; the peak is at the same location as in Fig. 4 butits height is smaller. In Fig. 7, G is 3py4; the peakhas a height and a width close to those of Fig. 4, butits location is changed, still corresponding to g 5 cos G.

From these and many other computations, the be-havior of the BSD’s can be summarized as follows:The peak is localized at g 5 cos G, its width increases

with s, and its height depends on G and decreaseswith m. These results agree with the theoreticalact that, for a plane wave, the BSD’s are proportionalo a Dirac distribution with support located at g 5 cos

G.20

Fig. 4. BSD’s versus g with s 5 1022, G 5 py4, and m 5 0.

Fig. 5. BSD’s versus g with the same parameters as in Fig. 4 butor s 5 5 3 1022.

Fig. 6. BSD’s versus g with the same parameters as in Fig. 4 butor m 5 100.


a

uq

tqTtmtd

fbIb~

mm

f

ws

2

These behaviors are also in agreement with a lo-calized interpretation. Furthermore, the behaviorof the BSD’s helps us to optimize the numerical com-putations, which can be carried out on a finite num-ber of m terms with quadratures on g carried out onlyin a finite interval centered on cos G. These resultsre used in Subsection 6.B.As far as the computational time is concerned, let

s remark that the localized approximation re-uires only one quadrature over z ~Subsection 3.B!,

which, furthermore, may sometimes be analyticallyevaluated @Eq. ~108!, for instance#. Conversely,he original formulation requires at least twouadratures @see Eqs. ~33! and ~34! of Ref. 11#.he localized approximation is therefore computa-ionally much more efficient, possibly by orders ofagnitude. It also stimulates a strong formal in-

erest insofar as it no longer requires the use ofistributions.

B. Incident Beam

The Poynting vector that corresponds to the incidentbeam can be computed by either of two methods:directly from the first-order Davis representation orby use of the Bromwich potentials @Eqs. ~7! and ~8!#rom which all electromagnetic field components cane derived @see Eqs. ~24!–~35! of Ref. 23, for example#.n the latter computations the BSD’s are evaluatedy use of the localized approximations @Eqs. ~29!, ~30!,64!, and ~87!#. The component Sz of the Poynting

vector, which corresponds to an incident beam de-fined by s 5 1022 and G 5 py4, is plotted in Fig. 8versus kz, with the computational method as a pa-rameter. The agreement between the two methodsis excellent, confirming the capability of the localizedapproximation to compute the BSD’s accurately.Other similar comparisons, with different parame-ters, lead to the same conclusion; for example, seeFig. 9 for s 5 0.1. Figures 10–13 provide comple-

entary views on the agreement between the twoethods. Figures 10 and 12 display the modulus of

Fig. 7. BSD’s versus g with the same parameters as in Fig. 4 butor G 5 3py4.


Fig. 8. Comparison of uSzu computed by summation ~and with thelocalized approximation! and from a first Davis beam with s 50.01.

Fig. 9. Comparison of uSzu computed by summation ~and with thelocalized approximation! and from a first Davis beam with s 5 0.1.

Fig. 10. Modulus of Hz versus kz computed by summation ~andith the localized approximation! and from a first Davis beam with5 0.01.

bp

Hz versus kz for s 5 0.01 and s 5 0.1. These twofigures can be directly compared with Figs. 8 and 9.Figures 11 and 13 display the phase of Hz versus kz.Let us remark that the phase essentially does notdepend on s. In these figures the range of kz haseen restricted because there is a fast evolution of thehase with kz.

7. Conclusion

A cylindrical localized approximation of the beam-shape distributions in the generalized Lorenz–Mietheory for cylinders has been introduced and rigor-ously justified for an arbitrary location and an arbi-trary orientation of the scatterer. As for thespherical localized approximation, this cylindrical lo-calized approximation, based on a first-order Davisbeam, anticipates the behavior of higher-order beamdescriptions.

Also, some numerical results have been pre-sented that confirm the ability of the cylindricallocalized approximation to be used in extensive nu-merical studies, both because it is accurate andbecause it allows one to speed up the evaluation ofBSD’s.

Appendix A

First we establish a simplified expression for the dis-tribution d~g 2 C!y~1 2 g2!. This relies on the fol-lowing property of Dirac distributions18,19,21:

Kd~g 2 C!

1 2 g2 , F~g!L 5 Kd~g 2 C!,F~g!

1 2 g2L5 F F~g!

1 2 g2Gg5C

5F~C!

1 2 C2 , (A1)

to be compared with

Kd~g 2 C!

1 2 C2 , F~g!L 51

1 2 C2 ^d~g 2 C!, F~g!&

51

1 2 C2 @F~g!#g5C 5F~C!

1 2 C2 , (A2)

which establishes that

d~g 2 C!

1 2 g2 5d~g 2 C!

1 2 C2 , (A3)

which is involved in Eq. ~44!.More generally, Eq. ~A1! is a special case of the

relation

Kdn~g 2 C!

1 2 g2 , F~g!L 5 ~21!nF ]n

]gn

F~g!

1 2 g2Gg5C

. (A4)

For n 5 1 we obtain

Kd9~g 2 C!

1 2 g2 , F~g!L 5 2F ]

]g

F~g!

1 2 g2Gg5C

5 2F9~C!

S2 2 F~C!2CS4 , (A5)

Fig. 11. Phase of Hz versus kz computed by summation ~and withthe localized approximation! and from a first Davis beam with s 50.01.

Fig. 12. Modulus of Hz versus kz computed by summation ~andwith the localized approximation! and from a first Davis beam withs 5 0.1.

Fig. 13. Phase of Hz versus kz computed by summation ~and withthe localized approximation! and from a first Davis beam with s 50.1.


l

L

sr

i

S

2

leading to Eq. ~49!. Case n 5 2 is processed simi-arly, as are higher-order cases.

Appendix B

In this appendix we establish that

I 5 *2`

1`

exp~2at2 1 ibt!dt 5 Îp

aexpS2b2

4a D (B1)

for any complex number a and b and for Re~a . 0!.et a be ~a 1 ib!. Then first we evaluate

I1 5 *2`

1`

exp~2at2!dt

5 *2`

1`

exp~2at2 2 ibt2!dt

5 *2`

1`

exp~2at2!@cos~bt2! 2 i sin~bt2!#dt, (B2)

which can be integrated with the aid of the symboliccomputation software Maple to yield

I1 5 S p

Îa2 1 b2D1y2HcosF12

arctanSbaDG

2 i sinF12

arctanSbaDGJ . (B3)

This expression can then be algebraically manipu-lated to yield

I1 5 S p

a 1 ibD1y2

5 Îp

a. (B4)

Next we consider the integral I of Eq. ~B1! with b 5c 1 id, which can be modified to yield

I 5 expS2b2

4a D *2`

1`

expH2~a 1 ib!Ft 2i~c 1 id!

2~a 1 ib!G2Jdt.

(B5)

We set

i~c 1 id!

2~a 1 ib!5 A 1 iB, (B6)

x 5 t 2 A, (B7)

o the integral on the right-hand side of Eq. ~B5!eads as

I2 5 *2`

1`

exp@2~a 1 ib!~x 2 iB!2#dx. (B8)

This integral is evaluated with a contour of integra-tion ~Ref. 24, pp. 113–115! made from C1, C2, C3, andC4, as shown in Fig. 14. Because there is no singu-


lar point inside the contour, we have

*C

exp@2~a 1 ib!z2#dz 5 S*C1

1 *C2

1 *C3

1 *C4

D3 exp@2~a 1 ib!z2#dz 5 0, (B9)

n which z is a complex variable.On C2 we have a bound according to

U*C2

exp@2~a 1 ib!z2#dzU5 U*

0

B

exp@2~a 1 ib!~R 1 iy!2#dyU5 Uexp@2~a 1 ib!R2# *

0

B

exp@2~b 2 ia!Ry#

3 exp@~a 1 ib!y2#dyU# uexp@2~a 1 ib!R2#iB exp~2ubuRB 1 aB2!u

5 B exp~2aR2 1 2ubuRB 1 aB2!, (B10)

where R is the fixed value of x for line C2. Takingthe limit R 3 `, we then have

limR3` *

C2

exp@2~a 1 ib!z2#dz 5 0. (B11)

imilarly, we establish that

limR3` *

C4

exp@2~a 1 ib!z2#dz 5 0. (B12)

Fig. 14. Integration contour.

Fx

s

9. J. A. Lock, “Scattering of a diagonally incident focused Gauss-

1

1

1

1

Next, using our result for I1, we have

limR3` *

C1

exp@2~a 1 ib!z2#dz

5 *2`

1`

exp@2~a 1 ib!x2#dx 5 Îpya. (B13)

inally, not forgetting that C3 runs from x 5 1R to5 2R, and using Eqs. ~B9! and ~B11!–~B13!, we

have

I2 5 2limR3` *

C3

exp@2~a 1 ib!z2#dz

5 limR3` *

C1

exp@2~a 1 ib!z2#dz 5 Îpya, (B14)

o we obtain

I 5 Îp

aexpS2b2

4a D . (B15)

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arbitrary-shaped beam,” Appl. Opt. 36, 4292–4304 ~1997!.2. N. Gauchet, T. Girasole, K. F. Ren, G. Grehan, and G. Goues-

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